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For pairs of spaces \n \n $${{\\,\\textrm{Lip}\\,}}_0(M)$$<\/jats:tex-math>\n \n \n \n \n \n Lip<\/mml:mtext>\n \n <\/mml:mrow>\n 0<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and C<\/jats:italic>(K<\/jats:italic>) we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. It is also showed that if a metric space M<\/jats:italic> contains a bilipschitz copy of the unit sphere \n \n $$S_{c_0}$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n \n c<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of the space \n \n $$c_0$$<\/jats:tex-math>\n \n \n c<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, then \n \n $${{\\,\\textrm{Lip}\\,}}_0(M)$$<\/jats:tex-math>\n \n \n \n \n \n Lip<\/mml:mtext>\n \n <\/mml:mrow>\n 0<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> admits a continuous operator onto \n \n $$\\ell _1$$<\/jats:tex-math>\n \n \n \u2113<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and hence onto \n \n $$c_0$$<\/jats:tex-math>\n \n \n c<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Using this, we provide several conditions for a space M<\/jats:italic> implying that \n \n $${{\\,\\textrm{Lip}\\,}}_0(M)$$<\/jats:tex-math>\n \n \n \n \n \n Lip<\/mml:mtext>\n \n <\/mml:mrow>\n 0<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is not a Grothendieck space. Finally, we obtain a new characterization of the Schur property for Lipschitz-free spaces: a space \n \n $$\\mathcal {F}(M)$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n (<\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> has the Schur property if and only if for every complete discrete metric space N<\/jats:italic> with cardinality d<\/jats:italic>(M<\/jats:italic>) the spaces \n \n $$\\mathcal {F}(M)$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n (<\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$\\mathcal {F}(N)$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n (<\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are weakly sequentially homeomorphic.\n<\/jats:p>","DOI":"10.1007\/s00025-024-02323-z","type":"journal-article","created":{"date-parts":[[2024,12,2]],"date-time":"2024-12-02T07:44:50Z","timestamp":1733125490000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Continuous Operators from Spaces of Lipschitz Functions"],"prefix":"10.1007","volume":"80","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8525-0532","authenticated-orcid":false,"given":"Christian","family":"Bargetz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jerzy","family":"K\u0105kol","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Damian","family":"Sobota","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,2]]},"reference":[{"key":"2323_CR1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-31557-7","volume-title":"Topics in Banach space theory","author":"F Albiac","year":"2016","unstructured":"Albiac, F., Kalton, N.J.: Topics in Banach space theory, 2nd edn. 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